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Delaunay Bifiltrations of Functions on Point Clouds
- Publication Year :
- 2023
-
Abstract
- The Delaunay filtration $\mathcal{D}_{\bullet}(X)$ of a point cloud $X\subset \mathbb{R}^d$ is a central tool of computational topology. Its use is justified by the topological equivalence of $\mathcal{D}_{\bullet}(X)$ and the offset (i.e., union-of-balls) filtration of $X$. Given a function $\gamma: X \to \mathbb{R}$, we introduce a Delaunay bifiltration $\mathcal{DC}_{\bullet}(\gamma)$ that satisfies an analogous topological equivalence, ensuring that $\mathcal{DC}_{\bullet}(\gamma)$ topologically encodes the offset filtrations of all sublevel sets of $\gamma$, as well as the topological relations between them. $\mathcal{DC}_{\bullet}(\gamma)$ is of size $O(|X|^{\lceil\frac{d+1}{2}\rceil})$, which for $d$ odd matches the worst-case size of $\mathcal{D}_{\bullet}(X)$. Adapting the Bowyer-Watson algorithm for computing Delaunay triangulations, we give a simple, practical algorithm to compute $\mathcal{DC}_{\bullet}(\gamma)$ in time $O(|X|^{\lceil \frac{d}{2}\rceil +1})$. Our implementation, based on CGAL, computes $\mathcal{DC}_{\bullet}(\gamma)$ with modest overhead compared to computing $\mathcal{D}_{\bullet}(X)$, and handles tens of thousands of points in $\mathbb{R}^3$ within seconds.<br />Comment: 28 pages, 7 figures, 8 tables. To appear in the proceedings of SODA24
- Subjects :
- Computer Science - Computational Geometry
Mathematics - Algebraic Topology
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2310.15902
- Document Type :
- Working Paper